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2013

Bekenstein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant

Ioannis Haranas Wilfrid Laurier University, [email protected]

Ioannis Gkigkitzis East Carolina University, [email protected]

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Recommended Citation Haranas, I., Gkigkitzis, I. Bekenstein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant. Modern Physics Letters A 28:19 (2013). DOI: 10.1142/S0217732313500776

This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. Bekestein Bound of Information Number N and its Relation to Cosmological Parameters in a Universe with and without Cosmological Constant

Ioannis Haranas1 Ioannis Gkigkitzis2

1 Department of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada E-mail:[email protected]

1 Departments of Mathematics, East Carolina University 124 Austin Building, East Fifth Street, Greenville NC 27858-4353, USA E-mail: [email protected] Abstract Bekenstein has obtained is an upper limit on the entropy S, and from that, an information number bound N is deduced. In other words, this is the information contained within a given finite region of space that includes a finite amount of energy. Similarly, this can be thought as the maximum amount of information required to perfectly describe a given physical system down to its quantum level. If the energy and the region of space are finite then the number of information N required in describing the physical system is also finite. In this short letter two information number bounds are derived and compared for two types of universe. First, a universe without a cosmological constant lamda and second a universe with a cosmological constant lamda  are investigated. This is achieved with the derivation of two different relations that connect the Hubble constant and cosmological constants to the number of information N. We find that the number of information N involved in a the two universes are identical or N  N , and 2 2  that the total mass of the universe scales as the square root of the information number N, containing. an information number N of the order of 10122. Finally, we expressed Calogero’s quantization action as a function of the number of information N. We also have found that in self-gravitating systems the number of information N in nats is the ratio of the total kinetic to total thermal energy of the system.

Key words: Bekenstein bound, cosmological constant, information, nats, entropy, mass of the universe, self-gravitating systems, Calogero's conjecture.

1. Introduction There is an upper bound for the ratio of the entropy S, to the energy E of a mass M, for any bounded system with an effective radius R. This is known as the Bekenstein (1981) upper bound given by the relation:

 k RE  S  2  B  . (1)  c 

Where  is Planck’s constant, and c the speed of light. Using Eq.(1) and the equation given in Haranas and Gkigkitzis (2013), for the entropy S to be (Lloyd, 2001):

S  k B N ln 2 , (2) ______In this paper both authors have contributed equally.

where kB is the Boltzmann constant Eqs. (1) and (2) we obtain that the number of information N in nats contained in the quantum states in the sphere is given by the equation:

2  RE  N    . (3) ln 2  c 

Where the number of information N is given in nats. Taking, R  c / H 0 to be the Hubble radius and assuming that the universe self gravitational energy E gr  H 0 (Johri, 1996), we find that the number of information in nats N is equal to:

2  E  2  E  2    2  R gr  N            u  , (4) ln 2  H  ln 2  E  ln 2  H  ln 2  2   0   gr   0   c u  Similarly Eq. (4) can be written as the ratio of the angular frequency of oscillation of the universe over its Hubble constant H, or the ratio of gravitational radius of the universe over the universe’s Compton wavelength. Next we find the entropy at the Hubble horizon to be:

2 k c 3 k c 5  c  S  B A  B   k   . (5) H H 2 B   4G  G H 0  H 0  P  With reference to Haranas and Gkigkitzis (2013) and equating Eq. (2) to (5) we obtain Hubble’s parameter as a function of the number of information N to be:

  c    H 0  , (6) ln 2     P N  Substituting Eq. (6) in (4) and given that N is positive we obtain that:

4  E 2  2  N 2   P  N . (7)  2 2  ln 2   c  Solving Eq. (7) for N we obtain that:

2 2 4  E 2 2  4  E  4  M  0  N   P       U  , (8)  2 2      ln 2   c  ln 2  E P  ln 2  m P 

c 5 c G where, and E P  is the Planck energy, m P  is the Planck mass, and  P  is the Planck G G c 3 length. Therefore we find that the number of information N at the horizon of the Friedman universe is just the square of ratio of the total energy of the universe that is attributed to the universe’s total mass to that of the Planck energy, or equivalently to the square of the total mass of the universe to that of the Planck mass. Similarly, in a universe with zero curvature and cosmological constant  and with reference to Haranas and Gkigkitzis (2013) we have that the cosmological constant can be written as a function of the information number bit N in the following way:

 3  3      max , (9)  N ln 2 2  ln 2 N  p 

2 3 where max  1 /  p  c / G (Haranas, 2002) the maximum cosmological constant, N the number of

information in nats,  p is the Planck length. Therefore writing the cosmological de-Sitter horizon as function the information N we obtain:

3 N ln 2 rH    P . (10) 1, 2   Substituting Eq. (10) into first part of Eq. (4) after simplifying in a way similar to Eq. (6) we obtain that:

4  E 2  2  N 2   P  N , (11)  2 2  ln 2   c  from which we obtain that:

2 2 4  E 2  2  4  E  4  M  0  N   P       U  , (12)   2 2      ln 2   c  ln 2  E P  ln 2  m P  and therefore we obtain that:

N  N , (13) 2 2  in other words the number of information contained in a Friedmann universe without cosmological constant results to a Bekenstein information bit bound that is identical to the universe involving a cosmological constant , or symbolically N  N  N . This is the number of information in nats with 2 F 2  the help of which information can be decompressed through matter and energy, for the two types of universe. We find that for both types of universe the number of information N depends on the same fundamental parameters i.e. the energy due to mass M over the Planck energy EP or equivalently the mass of the universe M over the Planck mass mP. From Eq. (4) we see that the number of information N constitutes the “connecting entity” of various fundamental cosmological parameters. As a result from Eq. (4) we obtain the gravitational radius of the universe can be expressed in the following way:

 2 Ec  R   U  . (14) grU    H 0 

Next, substituting for the universe’s Compton wavelength    / M c , and R  2GM / c 2 and cU U gr U

2 E  M U c we obtain that the mass of the universe is given by: c 3 M U  . (15) GH 0 This is an equation similar to that predicted by Hoyle (1958), Narlikar (1993), Carvalho (1995), Haranas and Harney (2009) as well as (Valev, 2010). For example Valev (2010) obtains the same equation via a dimensional analysis. Having expressed H0 as a function of the number of information N in nats, we can use Eq. (6) to express the mass of the universe as a function of the number of information N in the universe without cosmological constant to be:

2 2 c  P ln 2 N c  P max  P  M U   0.470 N  0.470 N  0.470 N . (16) G  G c c P

Finally, using that the Planck momentum can be written as m P c   /  P from the last expression of Eq. (18) we can obtain that:

M U  0.470 m P N . (17) Similarly, in the universe with cosmological constant  we obtain that:

2 c 2 c 3 ln 2 p max  P  M U   N  0.470 N  0.470 N  0.470 m P N (18) G   G c c P The first tem in the RHS of Eq. (18) agrees with the Eq. (8) given in Capozziello and Funkhouser (2009) and Eq. (3.8) in Funkhouser (2008). We express this equation as a function of information N, from which the number of information N in nats related to the universe’s total mass can be calculated. Therefore

53 8 taking M U  1 10 kg, (Immerman, 2001) and m P  2.176  10 kg obtain that:

2  M   U  122 0  N  4.526    0.996  10 nats. (19)  m P  The number 10122 appears in an ensemble of pure numbers naturally produced from fundamental cosmological parameters that might constitute a new-large number coincidence similar to that postulated by Dirac (1973). These numbers constitute a compelling, new large number coincidence problem (Funkhouser, 2008) and (Haranas and Gkigkitzis, 2013). This is possible after the derivation of two relations which connect the cosmological constant  and the Hubble constant H0 to the information number N. We find that the total mass of the universe has a N dependence on the information number bit N. Furthermore, we note that the mass of the universe expressed in terms of fundamental parameters that basically become coefficients of N term and carry units of mass. We see that for N =1 the universe achieves a mass M U  0.470 m P , for which Eq. (22) gives N = 0.999  1.0. Inflationary cosmological scenarios tell us that at time t  t P then m  m P , (Linde, 1990) and therefore we can say that a universe with M U  0.470 m P will imply that the first nat of information is decompressed through matter and energy at a time t earlier that the Planck time. In Capozziello et al., (2001) the authors examine self-gravitating systems, where they consider virialized systems, giving the equation for the total energy of the system to be:

2 E kin  U  0 , (20) where Ekin is the kinetic energy, and U the gravitational energy. Therefore, the total energy E given by Capozziello, (2001) is given by:

2 E  E kin  N 0 mv . (21) Using Capozziello’s equation for the total energy, we write the entropy of the self-gravitating system to be:

N mv 2 S  0 , (22) T and using Eq. (2) and (22) we obtain the number of information in nats related to the self-gravitating system in the following way:

N mv 2 N  E  2 E N  0  2 0  kin   kin   , (23) Tk B ln 2 ln 2  E th  TΔS where N0 is the total numbers of bodies of mass m contained within the self-gravitating system, and Eth is the total thermal energy of the system, and ΔS  k B ln 2 is the change in entropy. Similarly, using Eq. (23) we obtain an expression for the characteristic (minimal) unit of action  = τ (Capozziello, et al., 2001) per granular component to be:

A A      . (24) 3 / 2 3 / 2  k BT   E   N ln 2 kin  2    N ln 2  mv   E   th   Finally, with reference to Calogero’s work on cosmic quantization (Calogero, 1997) the author predicts a quantum of action h as a function of basic parameters of physics, namely:

h  G 1 / 2 m 3 / 2 R 1 / 2 , (25) where G is the gravitational constant, m mass of the nucleon or hydrogen atom, and R is the radius of the universe (Calogero, 1997). We can show the relation of Calogero’s quantum of action h, to number of information N in nats by taking the radius of the universe at the Hubble horizon in two kinds of different universes. First, at Hubble horizon of a Friedmann universe we have that R  c / H 0 and therefore we obtain that: 1 / 2  c  1 / 2 3 / 2   h  G m   (26)  H 0  Upon substituting Eq. (6) into Eq. (26) and eliminating the Planck length we obtain that:

1 / 4 1 / 4 1 / 4  ln 2  1 / 2  ln 2  1 / 2  Gh  h  Gm 3 N 1 / 4  Gm 3 N 1 / 4    p       3  , (27)        2 c  solving for h we obtain the following real solutions

h1  h  0 , (28)

1 / 3 2  ln 2   Gm  h  h  0.466   N 1 / 3 . (29) 2      2   c  Finally, in a de-Sitter universe with cosmological constant we have that Calogero’s equation becomes:

1 / 2 1 / 2 1 / 2  3   N ln 2  1 / 2 3 / 2   1 / 2 3 / 2   h  G m  G m  P    , (30)           thus eliminating the Planck length  P and solving for h we obtain the following real values for the quantum of action: , (31)

1 / 3 2  ln 2   Gm  h  0.466   N 1 / 3 . (32)      2   c  Using Eq. (29) and solving for N we obtain that:

2 3 5 3 5 6 2  c  3 16  c  16  m  N   h      P  , (33)  3 6  2 ln 2  G m  ln 2  Gm  ln 2  m  Therefore, the number of information N in Calogero’s scheme of cosmic quantization depends upon the ratio of two fundamental masses i.e. the Planck mass and mass of the nucleon raised to the sixth power. Similarly, we can say that Calogero’s quantum of action h as related to the universe’s radius calculated at the Hubble radius, involves an N 1 / 3 dependence on the number of information in nats. Using

8 27 m P  2.176  10 kg and m  1.670  10 kg (Calogero, 1997) we obtain that the number of information in nats is:

N  3.457  10 118 , (34) 10118 is a large number that appears in an ensemble of pure numbers that they are naturally produced from fundamental cosmological parameters, that they are probably part of a new-large number hypothesis, similar to the one postulated by Dirac. In the case where m  m P we obtain that: 16 5 N   7064 . (35) ln 2 Similarly, in the case of a self-gravitating system, let us consider the typical open star cluster Pleiades for

2 2 which M = 300Msolar R= 3.5 pc, N 0  300 stars and v  4.63  10 N 0 m / R km/s (Chandrasekhar,

1960), we obtain that v  0.43 km/s, with temperatures in the range 4000 K  T  15000 K. Using an average temperature T = 10000 K, we obtain that the number of information N, is:

N  1.154  10 57 . (36)

2. Conclusions In this letter we have used Bekenstein’s upper bound of entropy, to calculate an upper bound for the number of information bits N in two different types of universe. We find that the number of information N in a universe without a cosmological constant is identical to the number of information N in a universe with a cosmological constant lamda. Furthermore, we find that the mass of the universe can be expressed in terms of many various cosmological parameters that basically become the coefficients of the N term

and have units of mass, thus the mass of the universe M U  m P N and that the universe achieves a mass

M U  0.470 m P if N =1. Finally, we have expressed Calogero’s quantization action as a function of the number of information N. We have found that the action h as related to the radius of the universe, when taken at the universe’s horizon, involves a large number of information in nats, that is the same in a universe with or without a cosmological constant. In relation to Capozziello’s results for self-gravitating systems, we have found that the number of information N in nats is the ratio of the total kinetic to total thermal energy of the system.

References Bekenstein, J. D., Physical Rev. D, 23 (2), 287, 1981. Calogero, F., Cosmic origin of quantization, Physics. Letters, A 228, 335-346, 1997 Capozziello, S., De Martino, De Siena, S. S.,and Illuminati, F., Non-Newtonian Gravity, Fluctuative Hypothesis and the Sizes of Astrophysical Structures, Mod.Phys.Lett. A16, 693, 2001. Capozziello, S., and Funkhouser, S., A scaling law for the cosmological constant from a stochastic model for cosmic structures Mod.Phys.Lett. A24:1121-1128, 2009. Carvalho J. C., Derivation of the Mass of the Observable Universe, Int. J. Theor. Phys., 34, 2507-2509, 1995. Chandrasekhar, S., Principles of Stellar Dynamics, Dover Publications, pp. 200-201, 1960. Dirac, P. A. M., Cosmological Models and the Large Numbers Hypothesis Proc. R. Soc. Lond. 333, 403, 1973. Funkhouser, S., A new large-number coincidence and a scaling law for the cosmological constant, Proc. R. Soc. A. 464, 1345-1353, 2008. Haranas, I., The Temperature of a Black Hole in a De-Sitter Space-Time, Rom. Astron. Journa l, Vo. 12 No. 2, 2002. Haranas, I., and Harney, M., The Mass of the Universe and Other Relations in the Idea of a Possible Cosmic Quantum Mechanics, Progress in Physics, vol. 2, pp.50-51, 2009. Haranas, I., Gkigkitzis, I., The Number of Information Bits Related to the Minimum Quantum and Gravitational Masses in a Vacuum Dominated Universe, Astrophys Space Sci., DOI 10.1007/s10509-013- 1434-1, 2013. Hoyle F., Proceedings of 11th Solvay Conference in Physics, The Structure and Evolution of the Universe, Edited by R. Stoops, Brussels, 1958. Immerman, N., Sacramento Peak: The Universe. University of Massachusetts Amherst. 21 May 2001. Johri, V. B., The Early Universe, ISTITUTO PER LA RICERCA DI BASE, ITALY, Castello Principe Pignatelli, 86075 Monteroduni (IS), Molise, Italy, HADRONIC PRESS, pp. 33, 1999. Lloyd, S., Computational capacity of the universe, arXiv: quant-ph/0110141v1 24 Oct 2001. Linde, A., Particle Physics and Inflationary Cosmology, Harwood Academic Publishers, 1990. Narlikar J. Introduction to Cosmology. Cambridge University Press, Cambridge, p. 272., 1993 Valev, D., Estimations of total mass and energy of the universe, arXiv: 1004.1035v1 [physics.gen-ph] 7 Apr 2010.